Model-Based Clustering using multi-allelic loci data with loci selection

We propose a Model-Based Clustering (MBC) method combined with loci selection using multi-allelic loci genetic data. The loci selection problem is regarded as a model selection problem and models in competition are compared with the Bayesian Information Criterion (BIC). The resulting procedure selects the subset of clustering loci, the number of clusters, estimates the proportion of each cluster and the allelic frequencies within each cluster. We prove that the selected model converges in probability to the true model under a single realistic assumption as the size of the sample tends to infinity. The proposed method named MixMoGenD (Mixture Model using Genetic Data) was implemented using c++ programming language. Numerical experiments on simulated data sets was conducted to highlight the interest of the proposed loci selection procedure.

💡 Research Summary

This paper addresses a fundamental problem in population genetics: the identification of genetically homogeneous sub‑populations (clusters) from multi‑allelic loci data. Traditional approaches such as Wright’s F‑statistics require a priori definitions of populations based on linguistic, cultural, or geographic criteria, which are often subjective and may miss cryptic structure. Recent model‑based clustering tools (e.g., STRUCTURE, BAPS) treat each individual as arising from a mixture of latent populations, but they rely on computationally intensive Markov‑chain Monte Carlo (MCMC) procedures and do not incorporate a systematic variable (locus) selection step. Consequently, when a data set contains many loci that are irrelevant or even detrimental to detecting structure, the performance of these methods can deteriorate.

The authors propose a new method, MixMoGenD (Mixture Model for Genetic Data), that simultaneously estimates three key quantities: (i) the number of clusters K, (ii) the subset S of loci that truly contribute to clustering, and (iii) the population‑specific allele frequencies and mixing proportions. The problem is cast as a model‑selection task: each pair (K, S) defines a parametric mixture model M(K,S) with parameters θ = (π, α, β). Here π denotes the mixing proportions, α the allele frequencies for loci in S, and β the common allele frequencies for loci in the complement Sᶜ (assumed to be identical across all populations). The authors assume Hardy–Weinberg equilibrium within populations, complete linkage equilibrium, and that non‑informative loci share the same distribution across populations.

Because the integrated likelihood P(x | K,S) is analytically intractable, the Bayesian Information Criterion (BIC) is used as an asymptotic approximation to the log‑marginal likelihood. For a given (K,S), the BIC is computed as
BIC(K,S) = 2 ∑{i=1}^n log P{K,S}(x_i | θ̂_{ML}) – d(K,S) log n,
where θ̂_{ML} is the maximum‑likelihood estimate obtained via the Expectation–Maximization (EM) algorithm, and d(K,S) is the dimension of the parameter space. The selected model (Ķₙ, Ŝₙ) maximizes BIC over all admissible K (1 … K_max) and all non‑empty subsets S of the L loci.

A full exhaustive search over 2^L – 1 possible subsets is computationally prohibitive. Therefore the authors adopt a two‑step nested procedure inspired by Maugis et al. (2007). First, for each fixed K, a backward stepwise search is performed on the set of loci: starting with S = {1,…,L}, the algorithm iteratively removes the locus whose exclusion yields the greatest BIC improvement (if any) and then checks whether any previously excluded locus, when re‑added, would improve BIC again. This “exclude‑then‑include” cycle continues until no further improvement is possible. Second, after obtaining the optimal S(K) for each K, the algorithm selects the K that gives the highest BIC. The backward approach is preferred over a forward search because it allows the model to capture interactions among loci before discarding any.

Parameter estimation within a given (K,S) follows the standard EM scheme. In the E‑step, posterior cluster probabilities τ_{ik}=P(z_i=k | x_i,θ^{(r)}) are computed using current estimates of π and α for the loci in S. In the M‑step, mixing proportions are updated as the average of τ_{ik} across individuals, and allele frequencies α_{k,l,j} are updated as the τ‑weighted proportion of allele j observed at locus l in cluster k. For loci in Sᶜ, the common frequencies β_{l,j} are simply the empirical allele frequencies across the whole sample, because under assumption (H3) they are identical across clusters.

The paper provides a rigorous consistency proof: under the three biological assumptions (Hardy–Weinberg, linkage equilibrium, and identical distribution of non‑informative loci) and a mild identifiability condition, the BIC‑based selection of (K,S) converges in probability to the true model (K₀,S₀) as the sample size n → ∞. This result guarantees that, asymptotically, the procedure will recover the correct number of populations and the exact set of discriminative loci.

Implementation details are also discussed. The core EM and BIC calculations are written in C++/C for speed, and an R interface is supplied to make the tool accessible to biologists. The authors release the source code, sample data sets, and simulation scripts freely.

Extensive simulation studies evaluate the method under varying numbers of clusters (K = 2–5), total loci (L = 10–30), proportions of informative versus non‑informative loci, and allele frequency heterogeneity. Results consistently show that MixMoGenD’s locus‑selection step dramatically improves both the accuracy of estimating K (measured by the proportion of correctly identified K) and the quality of individual assignments (measured by Adjusted Rand Index) compared with using all loci indiscriminately. Moreover, because the EM algorithm replaces MCMC, computational times are substantially reduced, making the approach scalable to larger genomic data sets.

In conclusion, MixMoGenD offers a statistically sound, computationally efficient framework for simultaneous clustering and variable (locus) selection in multi‑allelic genetic data. It bridges a gap between model‑based population inference and modern high‑throughput genotyping, providing a practical tool for detecting cryptic population structure. Future work suggested by the authors includes extending the model to accommodate linkage disequilibrium, handling missing data, and applying the method to real whole‑genome data sets such as the 1000 Genomes Project.